Study on the arithmetic discontinuous groups

这个项目的目标之一是明确地确定所有签名为（O;e_1,e_2,e_3,e_4）的算术Fuchsian群Γ。(1)设K为n次的全实数代数域K。设a为K上的一个四元数代数，使得a【叉乘】_Q R = M_2(R)【对称】H^<n-1>，其中H为哈密顿四元数代数。设0是A的一个阶，设Γ^1（A，O）是由O（范数1）的单位群O^1衍生出来的一个Fuchsian群。如果一个Fuchsian群Γ与Γ^1 （a，O）可通约，则Γ称为k算术Fuchsian群。如果存在一个签名为（0;e_1,e_2,e_3,e_4）的K-算术Fuchsian群Γ，则度[K:Q]【小于等于】10。并给出了这类域K的判别式d(K)的显式上界D_0。我们确定了具有最小判别式d(K)的10次非基域K。(2)我们研究了K=Q的情况。设A是Q上的四元数代数，设O(f)是A中具有无平方层f的Eichler阶。设Γ^*（A，O(f)）是SL_2(R)中的归一化器ofΓ^1（A，O(f)）。我们证明了Γ是一个q算术的Fuchsian群。因此，我们可以显式地确定所有签名为（O;e_1,e_2,e_3,e_4）的q -算术Fuchsian群Γ。

One of the aims of this project is to determine all arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitely.(1)Let K be a totally real algbraic number field K of dgree n. Let A be a quaternion algebra over K such that A【cross product】_Q R〓M_2(R) 【symmetry】H^<n-1>, where H is the Hamilton quaternion algebra. Let O is an order of A. Let Γ^1(A,O) be a Fuchsian group derived from unit group O^1 of O of norm 1. If a Fuchsian group Γ is commensurable with Γ^1 (A,O), Γ is called K-arithmetic Fuchsian group. If there exists a K-arithmetic Fuchsian group Γ with signature (O;e_1,e_2,e_3,e_4), then the degree [K:Q]【less than or equal】10. Moreover, the explicit upper bound D_0 of the discriminant d(K) of such fields K is given. We have determined the imprimitive field K of degree 10 with minimum discriminant d(K). (c.f.K.Takeuchi [1])(2)We have studied the case K=Q. Let A be a quaternion algebra over Q and let O(f) be an Eichler order in A with square-free level f. Let Γ^*(A,O(f)) be the normalizer ofΓ^1(A,O(f)) in SL_2(R). We show that if Γ is a Q-arithmetic Fuchsian group. Then Γ is a subgroup of Γ^*(A,O) of finite index, (c.f.K.Takeuchi [2])Consequently, we can deternime all Q-arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitly.